What we are interested in here is a simpler situation, namely a many body quantum system
with nearest neighbour interactions, in the thermodynamic limit,
and we will mostly focus on the transverse field XY model. The notion of complexity
is easier to understand here, and provides important physical insights.
In quantum information theory, one quantification of complexity refers
to a minimal number of universal or elementary gates required to
construct a unitary transformation \(U\) , which acts on a given reference state \(\mathinner {|{\Psi _{R}}\rangle }\) 
to produce a given target state \(\mathinner {|{\Psi _{T}}\rangle }\) . Thus, with a few arbitrary choices (such as simple reference states),
the NC is defined as the number of elementary gates in the optimal circuit. A natural difficulty with this
quantification arises, as the choices of good reference states and elementary gates have a theoretically
infinite number of possibilities. Nielsen and collaborators [1]}, [2]}, [3]} therefore
adapted a geometric method for the evaluation of complexity, dubbed the Nielsen complexity (NC) or the
circuit complexity.
This geometric approach (explored in detail recently in [4]}, [5]}, [6]}, [7]})
involves a continuum of the unitary transformations \(U(\tau )\)  generated by any
(possibly) time-dependent Hamiltonian \(H(t)\) , with \(\tau \)  parametrising a path in the Hilbert space.
